Largest bulk gap of the complex Ginibre ensemble

The complex Ginibre ensemble is a non-Hermitian random matrix model. Introduced by Ginibre in 1965 [12], it consists of matrices with independent centered complex Gaussian entries of variance $1/n$. We refer to the monograph of Byun and Forrester [3] for an account of recent developments on the subject.

The unordered complex eigenvalues $z_{1},\ldots,z_{n}$ have joint probability measure

$$\frac{n^{n(n+1)/2}}{\pi^{n}\prod_{j=1}^{n}j!}\exp\!\left(-n\sum_{j=1}^{n}|z_{j}|^{2}\right)\prod_{1\leq i<j\leq n}|z_{i}-z_{j}|^{2}\prod_{j=1}^{n}\,\mathrm{d}^{2}z_{j}.$$

(1)

As is well-known, the associated point process is determinantal, and as $n\to\infty$, the empirical eigenvalue measure converges to the uniform probability measure on the unit disk. Thus a typical bulk nearest-neighbour distance is of order $n^{-1/2}$. The present paper concerns a substantially more delicate question: how large can such a distance be among all eigenvalues lying in a prescribed portion of the bulk?

The smallest gaps between nearest-neighbour eigenvalues arise from exceptionally close pairs and therefore require detailed knowledge of the asymptotic behavior of the correlation kernel. The largest gaps, by contrast, are created by exceptionally isolated eigenvalues and are governed by a large-deviation event: a disk centred at such an eigenvalue, with radius slightly larger than $n^{-1/2}$, must contain no other eigenvalue. The precise large-$n$ asymptotics of this hole probability were recently obtained in [5], and we will rely on this result.

The study of extreme eigenvalue spacings was initiated by Vinson [22] and now has a substantial history, particularly in dimension one. For the CUE and GUE, Ben Arous and Bourgade [2] obtained the joint limiting laws of the smallest gaps and the leading-order term of the largest gaps. Soshnikov’s earlier work [21] established Poisson statistics for the smallest spacings of a broad class of translation-invariant determinantal processes. The largest gaps were subsequently resolved at the fluctuation scale by Feng and Wei [9]: after a precise centering, the largest gaps of the CUE and GUE form a Poisson process and the ordered gaps have generalized Gumbel laws. More recently, Charlier [8] obtained sharp largest-gap asymptotics for general unitary-invariant Hermitian ensembles.

In dimension two, the theory is much younger. Shi and Jiang [19] proved that the smallest gaps of the complex Ginibre ensemble are of order $n^{-3/4}$ and converge, after rescaling, to Poisson statistics. This result was extended by Charlier [7] to general random normal matrices. The same scale and limiting Poisson statistics were obtained by Lopatto and Meeker [16] for the non-real bulk eigenvalues of the real Ginibre ensemble. Related Poisson limits for the smallest distances between zeros of Gaussian analytic functions on compact Riemann surfaces, together with an analogue for the planar Gaussian entire function, were established by Feng and Yao [10]. On the opposite side of the spacing spectrum, Otto [18] proved a Poisson limit for points with unusually large nearest-neighbour distance in the infinite Ginibre process observed through expanding windows, and deduced the leading scale of the maximum. Lopatto and Otto [17] then initiated the study of the largest bulk nearest-neighbour gap in the complex Ginibre ensemble. Writing $M_{n}(B)$ for the largest nearest-neighbour distance among eigenvalues anchored over a fixed bulk set $B$, as made precise in (5), they proved that

$$\frac{\sqrt{n}M_{n}(B)}{(4\log n)^{1/4}}\longrightarrow 1\qquad\text{in probability}.$$

(2)

Their work also provided a source of inspiration for the present work.

As mentioned earlier, the smallest and largest nearest-neighbour distances in the Ginibre ensemble (1) occupy the very different scales

$$n^{-3/4}\qquad\hbox{and}\qquad n^{-1/2}(4\log n)^{1/4},$$

respectively, and obtaining precise information on the fluctuations of the largest gap requires asymptotics for the hole probability on a disk of radius larger than $n^{-1/2}$.

Disk-hole probabilities for the Ginibre ensemble have a long history. Early contributions to this problem include Forrester [11] and Jancovici, Lebowitz and Manificat [14]. Adhikari and Reddy [1] and Charlier [6] studied hole probabilities for general shapes at leading order. For the much finer precision required here, Charlier’s analysis of disk counting probabilities [5] provides the precise expansion, including the order-one term, that underlies the analytic part of the present work: as $n\to\infty$ with $r$ fixed, he obtained

$\displaystyle\mathbb{P}(\#\{|z_{j}|<r\}=0)=\exp\left(C_{1}n^{2}+C_{2}n\log n+C_{3}n+C_{4}\sqrt{n}+C_{5}\log n+C_{6}+O(n^{-1/12})\right),$

(3)

where

		$\displaystyle C_{1}=-\frac{r^{4}}{4},\quad C_{2}=-\frac{r^{2}}{2},\quad C_{3}=r^{2}\left(1-\log(r\sqrt{2\pi})\right),$
		$\displaystyle C_{4}=a_{1}r,\quad C_{5}=\frac{1}{3},\quad C_{6}=a_{0}+\frac{2}{3}\log r,$

and $a_{1},a_{0}$ are defined in (9) and (12). We also mention that a result analogous to (3) for the spherical ensemble is available in [4].

Our aim is to pass from the leading law (2) to a complete extreme-value theorem. We identify the canonical tail transform, prove a Gumbel limit for the largest gap, and determine every deterministic term in the fourth-power centering down to order one. We also obtain the limiting laws of all fixed extreme order statistics. The answer is most naturally stated in terms of the reduced-Palm hole probability of the infinite Ginibre process. We write $\xi_{\infty}^{0,!}$ for the process governed by the reduced Palm distribution at the origin, with the distinguished Palm point removed. If a point is fixed at the centre of a disk, the microscopic event that its nearest neighbour lies farther than $R$ is

$$q(R)=\mathbb{P}\{\xi_{\infty}^{0,!}(D_{R})=0\}.$$

Consequently the correct analogue of the exponential tail transform for independent random variables is $\Psi(R)=-\log q(R)$. At a physical radius $r$, the expected number of anchors with an empty surrounding disk is asymptotic to

$$\frac{n|B|}{\pi}q(\sqrt{n}\,r).$$

Choosing $r$ by $\Psi(\sqrt{n}\,r)=\log(n|B|/\pi)+x$ therefore makes this mean tend to $e^{-x}$, suggesting both the Poisson law for exceedances and the Gumbel law for the maximum. Theorem 1.1 makes this heuristic exact.

Two features are specific to the required precision. First, the disk is anchored at an eigenvalue. The relevant quantity is therefore the reduced-Palm hole probability $q$, rather than the ordinary Ginibre hole probability $H$. For Ginibre the exact identity

$$q(R)=e^{R^{2}}H(R)$$

shows that confusing the two leaves the leading $R^{4}/4$ term unchanged but alters the entire $R^{2}$ correction. It is thus invisible in (2) and fatal at Gumbel accuracy. Second, one needs the large-hole expansion with an $o(1)$ error after inserting the growing microscopic radius $R\asymp(\log n)^{1/4}$. A fixed-parameter asymptotic cannot simply be evaluated along this regime. We derive the necessary compact-uniform form of Charlier’s disk-hole analysis [5], and then compare the finite and infinite reduced-Palm determinants with exponentially small error.

The result exhibits a useful dimensional contrast. In the sine process, the logarithm of a large interval probability is quadratic in its length; this produces largest one-dimensional bulk gaps on the scale $\sqrt{\log n}$ times the typical spacing. For the infinite Ginibre process, the logarithm of a large disk probability begins with $-R^{4}/4$; the maximal nearest-neighbour distance is therefore of order $(\log n)^{1/4}$ times the typical two-dimensional spacing. Beyond this leading exponent, the terms $R^{2}\log R$, $R^{2}$, $R$, $\log R$, and the constant term all survive, after inversion, at successively smaller but still deterministic orders. Keeping this entire hierarchy is what yields the explicit centering in Theorem 1.3.

Let us summarize the main contributions of the paper. First, we express the limiting law in the exact tail coordinate $\Psi$, which removes all ambiguity from the normalization. Second, we prove a sharp expansion of the Ginibre reduced-Palm hole probability, including explicit convergent integral formulae for the linear and constant coefficients. Third, we transfer the adaptive-radius Poisson approximation of [17] to the single deterministic radius selected by $q$, uniformly throughout a fixed bulk set. Finally, we invert the hole expansion and obtain a fourth-power normalization whose deterministic centering is accurate through order one.

The exact tail coordinate also clarifies which part of the argument is probabilistic and which part is analytic. The probabilistic input asserts that eigenvalues isolated at an adaptively chosen radius form, asymptotically, a Poisson process. The analytic input identifies that adaptive radius with a single deterministic radius, uniformly over the prescribed bulk set, and then evaluates it sharply. The latter step requires substantially more than the leading large-hole rate. An error of size $o(R^{2})$, for example, is enough to recover (2) but is much too large to locate a Gumbel fluctuation, whose change in the logarithmic tail is of order one. Our expansion retains all terms that remain visible after this inversion.

The paper is organized as follows. The remainder of this section fixes the normalization and states the principal results. Section 2 recalls the finite and infinite Ginibre kernels, identifies the reduced-Palm product, and records the Poisson input. Section 3 proves the full reduced-Palm hole expansion. Section 4 compares the finite and infinite reduced-Palm determinants uniformly for physical radii $r=O(n^{-1/2}(\log n)^{1/4})$. Section 5 passes from the adaptive Poisson theorem to a common deterministic threshold and inverts the tail asymptotics. The compact-uniform incomplete-gamma summation is proved in Appendix A.

Let $X_{n}=(x_{ij})_{i,j=1}^{n}$, where the $x_{ij}$ are independent centered complex Gaussian variables of variance $1/n$, each with density $(n/\pi)e^{-n|z|^{2}}\,\mathrm{d}^{2}z$, and write $\xi_{n}$ for its eigenvalue point process. Thus the circular-law droplet is the unit disk, and the bulk intensity in these coordinates is asymptotic to $n/\pi$. For $a\in\mathbb{C}$ and $r\geq 0$, write

$$D_{r}(a)=\{z\in\mathbb{C}:|z-a|<r\},\qquad D_{r}=D_{r}(0),$$

and write $|A|$ for the planar Lebesgue measure of a Borel set $A$. Throughout, $B\subset\mathbb{C}$ is a fixed bounded Borel set such that

$$0<|B|<\infty,\qquad|\partial B|=0,\qquad\overline{B}\subset\{z:|z|<1\}.$$

(4)

Define the anchored nearest-neighbour gaps and their maximum by

$$d_{n}(z)=\min_{w\in\xi_{n}\setminus\{z\}}|z-w|,\qquad M_{n}(B)=\max_{z\in\xi_{n}\cap B}d_{n}(z),$$

(5)

with the maximum over the empty set defined as $0$. Neighbours are taken from the full spectrum, not merely from $B$.

Let $\xi_{\infty}$ denote the infinite Ginibre process of intensity $1/\pi$, and let $\xi_{\infty}^{0,!}$ denote its reduced Palm process at the origin, with the distinguished Palm point removed. The probability that a point fixed at the origin has no other point within distance $r$ is

$$q(r):=\mathbb{P}\{\xi_{\infty}^{0,!}(D_{r})=0\}=\prod_{k=2}^{\infty}\frac{\Gamma(k,r^{2})}{\Gamma(k)},\qquad\Gamma(k,x)=\int_{x}^{\infty}t^{k-1}e^{-t}\,\mathrm{d}t.$$

(6)

The product converges locally uniformly, and $q$ is continuous and strictly decreasing from $1$ to $0$; see Lemma 2.2. Consequently

$$\Psi(r):=-\log q(r),\qquad r\geq 0,$$

is a continuous strictly increasing bijection of $[0,\infty)$ onto itself.

To extract an explicit centering from Theorem 1.1, we first determine the reduced-Palm hole probability sharply. Set

	$\displaystyle a_{2}$	$\displaystyle=2-\frac{1}{2}\log(2\pi),$		(8)
	$\displaystyle a_{1}$	$\displaystyle=\sqrt{2}\left\{\int_{-\infty}^{0}\log\!\left(\frac{\operatorname{erfc}y}{2}\right)\,\mathrm{d}y+\int_{0}^{\infty}\left[\log\!\left(\frac{\operatorname{erfc}y}{2}\right)+y^{2}+\log y+\log(2\sqrt{\pi})\right]\,\mathrm{d}y\right\},$		(9)

and define

	$\displaystyle J_{-}={}$	$\displaystyle\int_{-\infty}^{0}\left[2y\log\!\left(\frac{\operatorname{erfc}y}{2}\right)+\frac{e^{-y^{2}}(1-5y^{2})}{3\sqrt{\pi}\,\operatorname{erfc}y}\right]\,\mathrm{d}y,$		(10)
	$\displaystyle J_{+}={}$	$\displaystyle\int_{0}^{\infty}\left[2y\log\!\left(\frac{\operatorname{erfc}y}{2}\right)+\frac{e^{-y^{2}}(1-5y^{2})}{3\sqrt{\pi}\,\operatorname{erfc}y}+\frac{11}{3}y^{3}+2y\log y+\left(\frac{1}{2}+2\log(2\sqrt{\pi})\right)y\right]\,\mathrm{d}y,$		(11)
	$\displaystyle a_{0}={}$	$\displaystyle\frac{1}{4}\log(2\pi)-\zeta^{\prime}(-1)-2J_{-}-2J_{+}.$		(12)

Here $\operatorname{erfc}y=(2/\sqrt{\pi})\int_{y}^{\infty}e^{-t^{2}}\,\mathrm{d}t$, and $\zeta$ denotes the Riemann zeta function. All four integrals converge absolutely after the displayed renormalizations. Numerical quadrature of these convergent integrals gives

$$a_{1}=-2.3014159133\ldots,\qquad a_{0}=1.0063109967\ldots.$$

Inverting (13) yields an explicit fourth-power centering. For all sufficiently large $n$, define

	$\displaystyle\beta_{n}={}$	$\displaystyle 4\log\frac{n|B|}{\pi}-4\sqrt{\log\frac{n|B|}{\pi}}\,\log\!\left(2\sqrt{\log\frac{n|B|}{\pi}}\right)+8a_{2}\sqrt{\log\frac{n|B|}{\pi}}$
		$\displaystyle+4\sqrt{2}\,a_{1}\left(\log\frac{n|B|}{\pi}\right)^{1/4}+2\left[\log\!\left(2\sqrt{\log\frac{n|B|}{\pi}}\right)\right]^{2}$
		$\displaystyle+\left(\frac{10}{3}-8a_{2}\right)\log\!\left(2\sqrt{\log\frac{n|B|}{\pi}}\right)+8a_{2}^{2}-4a_{2}+4a_{0}.$		(14)

The square of the microscopic centering radius $\beta_{n}^{1/4}$ for $\sqrt{n}\,M_{n}(B)$ in (16) has the simpler large-$n$ expansion

$$\beta_{n}^{1/2}=2\sqrt{\log\frac{n|B|}{\pi}}-\frac{1}{2}\log\log\frac{n|B|}{\pi}+4-\log(4\pi)+o(1).$$

Consequently the largest gap has the readable location formula

$$nM_{n}(B)^{2}=2\sqrt{\log\frac{n|B|}{\pi}}-\frac{1}{2}\log\log\frac{n|B|}{\pi}+4-\log(4\pi)+o_{\mathbb{P}}(1).$$

(17)

Formula (14) retains every deterministic correction through order one. In decreasing order, their sizes are

$$\sqrt{\log n}\log\log n,\quad\sqrt{\log n},\quad(\log n)^{1/4},\quad(\log\log n)^{2},\quad\log\log n,\quad 1.$$

We collect the determinantal facts used later and keep track of the normalization. If $X_{n}$ has density $(n/\pi)^{n^{2}}e^{-n\operatorname{Tr}X_{n}X_{n}^{*}}$ on $\mathbb{C}^{n\times n}$, the complex Schur decomposition, followed by integration over the strictly upper triangular entries and the unitary factor, gives (1); see [3, Chapter 2]. Since

$$\prod_{1\leq i<j\leq n}|z_{i}-z_{j}|^{2}=\left|\det[z_{i}^{j-1}]_{i,j=1}^{n}\right|^{2},$$

the density is an orthogonal-polynomial ensemble for the planar Gaussian weight. The monomials satisfy

$$\int_{\mathbb{C}}z^{j}\bar{z}^{k}e^{-n|z|^{2}}\,\mathrm{d}^{2}z=\frac{\pi j!}{n^{j+1}}\,\mathbf{1}_{\{j=k\}},$$

(18)

as follows by passing to polar coordinates. Thus the functions $\varphi_{j,n}$ below form an orthonormal family in $L^{2}(\mathbb{C},\,\mathrm{d}^{2}z)$.

For completeness, recall that the $k$-point correlation function $\rho_{n}^{(k)}$ is characterized by

$\displaystyle\mathbb{E}\!\sum_{z_{1},\ldots,z_{k}\in\xi_{n}}^{\neq}f(z_{1},\ldots,z_{k})=\int_{\mathbb{C}^{k}}f(z_{1},\ldots,z_{k})\rho_{n}^{(k)}(z_{1},\ldots,z_{k})\prod_{j=1}^{k}\,\mathrm{d}^{2}z_{j}$

(19)

for every nonnegative measurable $f$, where the sum is over ordered tuples of distinct points. Andréief’s identity applied to (1) and (18) yields

$$\rho_{n}^{(k)}(z_{1},\ldots,z_{k})=\det[K_{n}(z_{i},z_{j})]_{i,j=1}^{k}.$$

With respect to planar Lebesgue measure, $\xi_{n}$ is determinantal with kernel [13, Theorem 4.3.10]

	$\displaystyle\varphi_{j,n}(z)$	$\displaystyle=\left(\frac{n^{j+1}}{\pi j!}\right)^{1/2}z^{j}e^{-n|z|^{2}/2},$		(20)
	$\displaystyle K_{n}(z,w)$	$\displaystyle=\sum_{j=0}^{n-1}\varphi_{j,n}(z)\overline{\varphi_{j,n}(w)}=\frac{n}{\pi}e^{-\frac{n}{2}(|z|^{2}+|w|^{2})}\sum_{j=0}^{n-1}\frac{(nz\bar{w})^{j}}{j!}.$

The microscopic infinite Ginibre kernel is the locally trace-class projection

$$K_{\infty}(z,w)=\sum_{j=0}^{\infty}\varphi_{j}(z)\overline{\varphi_{j}(w)}=\frac{1}{\pi}e^{z\bar{w}-|z|^{2}/2-|w|^{2}/2}.$$

(21)

Here $\varphi_{j}(z)=z^{j}e^{-|z|^{2}/2}/\sqrt{\pi j!}$, and $\xi_{\infty}$ denotes the determinantal process with kernel $K_{\infty}$. Its unit-disk-scale version, of intensity $n/\pi$, is

$$K_{\infty,n}(z,w)=nK_{\infty}(\sqrt{n}\,z,\sqrt{n}\,w)=\frac{n}{\pi}e^{nz\bar{w}-n|z|^{2}/2-n|w|^{2}/2}.$$

(22)

The diagonal of the finite kernel has the useful Poisson representation

$$K_{n}(z,z)=\frac{n}{\pi}e^{-n|z|^{2}}\sum_{j=0}^{n-1}\frac{(n|z|^{2})^{j}}{j!}=\frac{n}{\pi}\mathbb{P}\{\operatorname{Pois}(n|z|^{2})\leq n-1\}.$$

(23)

In particular, if $|z|\leq s$ with fixed $s<1$, a Chernoff bound gives

$$K_{n}(z,z)=\frac{n}{\pi}\bigl(1+O_{s}(e^{-c_{s}n})\bigr).$$

(24)

This is the quantitative bulk flatness that later makes the deterministic threshold independent of the location of the anchor.

Although the expression (21) is not literally translation invariant, its associated point process is. Indeed, for $a\in\mathbb{C}$, the magnetic translation

$$(U_{a}f)(z)=e^{i\operatorname{Im}(z\bar{a})}f(z-a)$$

is unitary on $L^{2}(\mathbb{C},\,\mathrm{d}^{2}z)$. The identity

$$K_{\infty}(z+a,w+a)=e^{i\operatorname{Im}(z\bar{a})}K_{\infty}(z,w)e^{-i\operatorname{Im}(w\bar{a})}$$

shows that it conjugates the compression of $K_{\infty}$ to $D_{r}(0)$ with that to $D_{r}(a)$. Fredholm determinants are invariant under this conjugation. Consequently both ordinary and reduced-Palm disk-hole probabilities depend only on the radius, once the Palm point and disk centre are translated together. The same statement holds for $K_{\infty,n}$, with magnetic phase $e^{in\operatorname{Im}(z\bar{a})}$. Dilation by $\sqrt{n}$ identifies its disk of physical radius $r$ with a disk of microscopic radius $\sqrt{n}\,r$ for $K_{\infty}$.

The rotational symmetry of Ginibre gives an additional exact structure that is particularly useful for disk events. The following result is usually referred to as Kostlan’s theorem [15]; see also [3, Section 2.9].

For $k\geq 1$ and $x\geq 0$, write

$$Q(k,x)=\frac{\Gamma(k,x)}{\Gamma(k)},\qquad\Gamma(k,x)=\int_{x}^{\infty}t^{k-1}e^{-t}\,\mathrm{d}t,qquad\gamma(k,x)=\int_{0}^{x}t^{k-1}e^{-t}\,\mathrm{d}t.$$

(25)

Kostlan’s representation gives a short alternative route to the ordinary disk product, but not directly to the anchored nearest-neighbour event. The latter is a statement under a reduced Palm measure, and the Palm conditioning changes precisely one radial mode. We therefore return to the operator formulation.

For any of the kernels $K_{n},K_{\infty,n},K_{\infty}$, the reduced-Palm kernel at $a\in\mathbb{C}$ is [20, Theorem 1.7]

$$K^{a}(z,w)=K(z,w)-\frac{K(z,a)K(a,w)}{K(a,a)}.$$

(27)

To make this projection statement precise, regard the kernels as operators on $L^{2}(\mathbb{C},\mathrm{d}^{2}z)$. The reproducing identity shows that (27) is the orthogonal projection onto the functions in the corresponding weighted Fock range that vanish at $a$. In particular,

$$\mathcal{H}_{n}=\operatorname{span}(\varphi_{0,n},\ldots,\varphi_{n-1,n}),\qquad\mathcal{H}_{\infty,n}=\overline{\operatorname{span}}\{\varphi_{j,n}:j\geq 0\},\qquad\mathcal{H}_{n}^{a}\subset\mathcal{H}_{\infty,n}^{a}.$$

Under the dilation $z\mapsto z/\sqrt{n}$, the intensity-$1/\pi$ process $\xi_{\infty}$ becomes the process with kernel $K_{\infty,n}$. Hence its reduced-Palm hole probability in a physical disk of radius $r$ is

$$\det(I-\mathbf{1}_{D_{r}(a)}K_{\infty,n}^{a}\mathbf{1}_{D_{r}(a)})=q(\sqrt{n}\,r)=\prod_{k=2}^{\infty}Q(k,nr^{2}).$$

(29)

Returning to the microscopic radius $R$, the omission of the $k=1$ factor in $q$ has a direct probabilistic meaning. Under the ordinary process, the constant Fock mode contributes the eigenvalue $1-e^{-R^{2}}$ to the disk compression. Conditioning on a point at the centre and then removing that point deletes exactly this mode. Thus the quotient of the two hole probabilities is the reciprocal of $Q(1,R^{2})$, which is $e^{R^{2}}$. This factor is negligible compared with the leading exponent $R^{4}/4$, but it is one of the dominant corrections at the fluctuation precision considered here.

We shall repeatedly use the Fredholm-determinant interpretation in the following elementary form. If $K$ is a locally trace-class Hermitian contraction defining a determinantal process $\eta$, then for every bounded Borel set $D$,

$$\mathbb{P}\{\eta(D)=0\}=\det(I-\mathbf{1}_{D}K\mathbf{1}_{D}).$$

(30)

Indeed, the number of points in $D$ is distributed as a sum of independent Bernoulli variables with parameters equal to the eigenvalues of the compression. Formula (30) follows by multiplying their zero probabilities. This also shows directly that enlarging $D$ decreases the void probability.

For a deterministic radius $r$, introduce the exceedance process

$$\mathcal{E}_{n}(r)=\sum_{a\in\xi_{n}\cap D_{1}}\mathbf{1}_{\{(\xi_{n}\setminus\{a\})(D_{r}(a))=0\}}\,\delta_{a}.$$

(33)

Thus $K_{n}(a,a)q_{n}(a,r)$ is the intensity density of anchors whose nearest-neighbour distance exceeds $r$. In the bulk, (24) and the finite/infinite comparison proved in Section 4 reduce it to $(n/\pi)q(\sqrt{n}\,r)$. This calculation explains both the factor $n|B|/\pi$ in Theorem 1.1 and why the reduced-Palm, rather than ordinary, hole probability is the relevant tail.

For a chosen target intensity $\kappa>0$, Lopatto and Otto select the radius separately at each anchor so that the integrand in (34) is exactly $\kappa$. The preceding monotonicity lemma guarantees that this adaptive radius is unambiguously defined. Indeed, uniformly for $|a|<1$,

$$K_{n}(a,a)\geq\frac{n}{\pi}\mathbb{P}\{\operatorname{Pois}(n)\leq n-1\}=\frac{n}{2\pi}(1+o(1)),$$

where the inequality follows from (23) and monotonicity of the Poisson distribution function. Thus $K_{n}(a,a)>\kappa$ for all large $n$; since $q_{n}(a,\,\cdot\,)$ decreases continuously from $1$ to $0$, the defining equation below has a unique solution.

We use the following theorem of Lopatto–Otto [17, Theorem 1].

To verify the normalization, let $\widetilde{K}_{n},\widetilde{q}_{n}$, and $\widetilde{R}_{n,\kappa}$ denote the quantities in [17, Theorem 1], which are written in variance-one coordinates. Under $z=\sqrt{n}\,a$,

$$K_{n}(a,a)=n\widetilde{K}_{n}(\sqrt{n}\,a,\sqrt{n}\,a),\qquad q_{n}(a,r)=\widetilde{q}_{n}(\sqrt{n}\,a,\sqrt{n}\,r),$$

and hence

$$R_{n,\kappa}(a)=n^{-1/2}\widetilde{R}_{n,\kappa}(\sqrt{n}\,a).$$

Thus $\Xi_{n,\kappa}$ is exactly the push-forward of the process in [17, Theorem 1] under $z\mapsto z/\sqrt{n}$, and its intensity equation is $K_{n}(a,a)q_{n}(a,R_{n,\kappa}(a))=\kappa$. The cited theorem uses closed rather than open disks. This does not change the process: by Campbell–Palm, the expected number of relevant pairs lying on $\partial D_{R_{n,\kappa}(a)}(a)$ is zero, since every reduced-Palm one-point intensity is absolutely continuous and every circle has zero planar area.